Game theory

The mathematics of strategic interaction: situations where your best move depends on what other people do, and theirs on yours. This sheet covers how to model a game, the solution concepts with worked examples, the canonical games and where they appear in business, startups, teams and politics, and the most useful practical lesson of all: when the equilibrium is bad, change the game. The oligopoly models (Cournot, Bertrand, Stackelberg) are in microeconomics; decisions against nature rather than against people are in decision-making and probability and risk; the feedback-loop view of the same traps (escalation, tragedy of the commons) is in systems thinking.

What a game is

A game is any situation where at least two decision makers affect each other's outcomes and know it. Chess, a price war, a salary negotiation, a nuclear stand-off and deciding who cleans the kitchen are all games.

Payoffs are von Neumann–Morgenstern utilities: numbers whose expected values rank risky prospects. That is what lets a player compare a 50% chance of 10 with a sure 4, and it is why mixed strategies make sense. They must include everything the player cares about (fairness, face, the future), or the analysis answers the wrong question.

The standard assumptions, and when they fail

A short history

Classifying games

Solution concepts

A solution concept is a rule for predicting what rational players will do. Stronger concepts make fewer predictions but rely on fewer assumptions.

Dominant strategies

Strategy $s$ strictly dominates $s'$ if it gives a strictly higher payoff whatever the others do. A rational player never plays a strictly dominated strategy, and if one strategy dominates all others, plays it without needing to predict anyone. The prisoner's dilemma is the famous case: defecting is dominant, yet both defecting is worse for both than both cooperating. Weak dominance (never worse, sometimes better) is a weaker argument: eliminating weakly dominated strategies can depend on the order of elimination.

Iterated elimination of strictly dominated strategies

If players know each other is rational, delete dominated strategies, then look again: new strategies may now be dominated. A game is dominance solvable if this leaves one outcome.

Row's best-looking payoffs (8 and 9) are in the column Column will never play. Each step needs one more layer of "I know that you know", which is why long elimination chains predict real behavior poorly.

Best responses and Nash equilibrium

A best response is a strategy that maximizes your payoff given the others' strategies. A Nash equilibrium is a profile where everyone is playing a best response to everyone else: nobody can gain by changing strategy alone.

Underline method (here: bold). For each column, mark Row's best payoff; for each row, mark Column's best payoff. Cells with both marked are pure-strategy Nash equilibria. Two firms choose a platform (standard A, standard B, or build their own):

Two equilibria, (A, A) and (B, B). The theory says the firms will coordinate on one standard; it does not say which. That selection problem is where focal points, history, commitment and first moves come in.

Mixed strategies and the indifference principle

A mixed strategy randomizes over actions. In a mixed equilibrium each player's mix makes the opponent indifferent between the actions the opponent uses; otherwise the opponent would play only the better one.

Matching pennies: both show heads or tails; Row wins 1 if they match, Column wins 1 if not.

No pure equilibrium: whoever is predictable loses. If Column plays heads with probability $q$, Row's expected payoff is $q - (1-q) = 2q - 1$ from heads and $1 - 2q$ from tails. Indifference gives $q = \tfrac12$; by symmetry Row also mixes 50/50, and the game's value is 0.

Penalty kicks. Chiappori, Levitt and Groseclose (2002) studied 459 penalties in the French and Italian top divisions. A simplified 2 × 2 version of their scoring rates (kicker's "natural" side is L, the left for a right-footer; "wrong side" lumps a goalkeeper staying in the middle with diving the wrong way):

Entries are the kicker's scoring probability; the goalie gets 1 minus it (constant-sum). With the goalie diving L with probability $g$, the kicker is indifferent when

With the kicker going L with probability $k$, the goalie is indifferent when $0.636k + 0.893(1-k) = 0.944k + 0.437(1-k)$, so $k = 0.456/0.764 \approx 0.60$. Goals are scored about 74% of the time. The counterintuitive prediction, confirmed in their data, is that goalies go to the kicker's natural side more often than kickers do (56.6% of goalie dives vs 44.9% of kicks, counting all kicks). They could not reject that scoring rates were equal across directions, as equilibrium requires, and found no evidence that players' choices depended on the opponent's move in the same kick.

Minimax and zero-sum games

In a two-person zero-sum game with mixed strategies, von Neumann's minimax theorem (1928) says

where $A$ is Row's payoff matrix. Maximizing your guaranteed payoff (maximin) and minimizing the opponent's best payoff (minimax) give the same value $v$ and the same strategies as Nash equilibrium. In the penalty game $v \approx 0.74$: the kicker can guarantee roughly 74% whatever the goalie does, and the goalie can hold it there. Zero-sum games are the one class where equilibrium play is also the safe play: being predictable is the only way to lose, so the practical rule is to randomize genuinely (use a die, not your gut, because people are bad at producing random sequences).

Efficiency versus equilibrium

The gap between the equilibrium and the efficient outcome is the cost of the game. Much of strategy, management and policy is closing it by changing the game, not by exhorting players to be nicer.

Focal points

When a game has several equilibria, people often coordinate on the one that stands out for cultural, historical or geometric reasons: a focal point or Schelling point. In The Strategy of Conflict (1960) Schelling asked people where and when they would try to meet a stranger in New York with no way to communicate; most of his informal sample chose noon, and a majority chose Grand Central Station (paraphrased from the book's report).

Use it: in negotiation, propose the focal point first (a clean number, an existing standard, a precedent that favors you). In coordination, make your preferred equilibrium the obvious one before others think about it.

Evolutionarily stable strategies

Maynard Smith and Price (1973) applied game theory to animals that do not reason at all: strategies spread when they earn more fitness. A strategy $s^*$ is an evolutionarily stable strategy (ESS) if a population playing it cannot be invaded by a small group of mutants playing $m$:

Hawk–dove. Contestants fight over a resource worth $V$; fighting costs the loser $C$. Hawks escalate; doves display and retreat.

If $V \ge C$, hawk is the ESS. If $V \lt C$, neither pure strategy is stable. With a share $p$ of hawks,

and setting them equal gives $p^* = V/C$. With $V = 4$ and $C = 10$: 40% hawks, and everyone averages $0.6 \times 2 = 1.2$, less than the 2 an all-dove population would get. Evolution, like rationality, does not guarantee efficiency. The same logic explains ritualised fights, costly territorial displays and why "limited war" conventions persist.

The canonical games

Payoffs below are (row, column), higher is better.

Prisoner's dilemma

The defining ordering is $T > R > P > S$ (temptation, reward, punishment, sucker); for repeated play one also requires $2R > T + S$, so taking turns exploiting each other is worse than steady cooperation. Defect is dominant; (defect, defect) is the unique equilibrium and is Pareto-dominated.

Stag hunt

After Rousseau's parable (Discourse on Inequality, 1755): hunt the stag together (big payoff, needs both) or a hare alone (small, safe).

Both (stag, stag) and (hare, hare) are equilibria. Stag is payoff dominant; hare is risk dominant (safe whatever the other does). Hunting stag pays only if you believe the other will with probability at least $p$, where $4p = 3$, so $p \ge 0.75$. Unlike the prisoner's dilemma, nobody is tempted to betray a cooperator: the only problem is doubt. Instances: adopting a new tool or process as a team, a coordinated rewrite, joining a new network, founders all quitting their jobs together. Escape: assurance: visible commitments, small reversible first steps, deposits, a credible leader who goes first.

Chicken and hawk–dove

Two drivers speed at each other; whoever swerves is "chicken".

Pure equilibria: one swerves, the other does not. In the symmetric mixed equilibrium each goes straight with probability $p$ where $-p = 1 - 11p$, so $p = 0.1$, and the crash happens 1% of the time. Chicken and hawk–dove are the same game. Instances: brinkmanship over deadlines and budgets, two companies both refusing to exit a market, standoffs in negotiations. The winning move is visible, irreversible commitment to straight before the other commits; the losing scenario is both committing.

Battle of the sexes (coordination with conflict)

Both want to be together but prefer different venues. Two pure equilibria; the mixed one (each goes to their own favorite with probability ⅔) gives each an expected ⅔, worse than either pure equilibrium. Instances: which standard two firms adopt (format wars), which language a merged codebase uses, whose process a merged team keeps. Resolution: move first and commit, set the focal point, alternate over time, or pay a side payment.

Ultimatum and dictator games

Ultimatum: the proposer offers a split of a sum; the responder accepts (split happens) or rejects (both get nothing). Backward induction: a money-maximizing responder accepts any positive amount, so the proposer offers the smallest unit. Evidence: Güth, Schmittberger and Schwarze (1982) ran the first experiment, and thousands of replications since show the same pattern: typical offers are 40–50%, the modal offer is an equal split, and acceptance falls quickly for offers below about 20% (Güth and Kocher's 2013 survey). Responders pay to punish what they see as unfair.

Dictator: the responder cannot reject. Self-interest predicts giving nothing; many people still give something (Kahneman, Knetsch and Thaler 1986), but Forsythe et al. (1994) found lower offers in the dictator game than in the ultimatum game, so part of ultimatum generosity is fear of rejection, not pure fairness. Giving varies a lot with framing, anonymity and social distance (Engel's 2011 meta-study of more than a hundred experiments).

Lesson for negotiators: a take-it-or-leave-it offer that looks exploitative gets rejected even when accepting is "rational". Leave the other side a split they can defend to themselves.

Public goods and the tragedy of the commons

Public goods game. Each of $n$ players has endowment $e$ and contributes $c_i$; the pot is multiplied by $m$ (with $1 \lt m \lt n$) and shared equally:

Each unit you contribute returns only $m/n \lt 1$ to you, so contributing nothing is dominant, yet everyone contributing everything is best for all. With $n = 4$, $e = 20$, $m = 1.6$: full contribution pays each 32; none pays 20. In experiments people contribute a fair amount at first and contributions decline with repetition; Fehr and Gächter (2000) found that with a costly option to punish free riders, near-complete cooperation can be reached and maintained.

Tragedy of the commons (Hardin, Science, 1968): each herder adds animals to a shared pasture because they get the full benefit of an extra animal but bear only a fraction of the damage. With value per animal $a - N$ (where $N$ is the total herd), cost $c$ per animal and $n$ herders, each choosing their own herd size gives

Hardin concluded that only privatisation or state control could save a commons. Elinor Ostrom (Governing the Commons, 1990) documented communities that had managed fisheries, forests and irrigation for centuries without either, and distilled eight design principles of long-enduring institutions:

Ostrom shared the 2009 economics prize. Hardin's "commons" was really an open-access resource; a commons with a defined community, rules and monitoring is a repeated game, not a one-shot dilemma.

Traveler's dilemma and the centipede game

Traveler's dilemma (Basu, American Economic Review, 1994). Two travelers each claim between 2 and 100 for identical lost antiques. Both receive the lower claim; the lower claimant gets 2 extra and the higher claimant 2 less. Undercutting the other by 1 always pays, so the unique equilibrium is (2, 2). Capra, Goeree, Gomez and Holt (1999) found claims near the top when the bonus and penalty were small and closer to 2 when they were large: the equilibrium predicts well only when deviating from it is costly.

Centipede (Rosenthal 1981). Two players alternately either take the larger share of a growing pot (ending the game) or pass (the pot grows). Backward induction from the last node says take at the first move, which gives both almost nothing. McKelvey and Palfrey (1992) found subjects rarely do; most pass several times. Lesson: long chains of backward induction require every player to trust every other player's rationality at every step. One doubt, and cooperating for a while becomes sensible.

Dollar auction

Shubik (1971): a dollar is auctioned to the highest bidder, but the second-highest bidder also pays their bid. Once two people have bid, each is always better off topping the other: at 95¢ vs 90¢ the trailer should bid $1.00 (losing 0 instead of 90¢); at $1.00 vs 95¢ the other should bid $1.05 (losing 5¢ instead of 95¢), and so on without limit. Each step is rational given sunk bids; the whole path is ruinous.

Instances: wars of attrition, bidding wars for acquisitions or talent, patent races, litigation, "we've invested too much to stop" projects. Defenses: set a walk-away limit before entering, treat sunk costs as sunk, and avoid all-pay contests unless you have a decisive advantage (see cognitive biases on sunk cost and escalation of commitment).

Keynesian beauty contest (guess ⅔ of the average)

Keynes (The General Theory, 1936, chapter 12) compared stock picking to a newspaper contest where you win by picking the faces others will pick. The p-beauty contest: everyone picks a number from 0 to 100; the winner is closest to $p$ times the average. With $p = \tfrac23$, iterated dominance drives the only equilibrium to 0:

Nagel (1995) found a first-round mean of about 37 with $p = \tfrac23$, consistent with most people doing one or two steps, and choices falling toward 0 over four rounds as players learned. Picking 0 in round one loses: the winning strategy is to be exactly one step deeper than the crowd. Markets, product launches and hype cycles often reward predicting beliefs about beliefs, not fundamentals.

Volunteer's dilemma

Diekmann (1985): someone must pay cost $c$ to produce a benefit $b$ for everyone (call the ambulance, fix the flaky test, report the bug). If nobody volunteers, everyone gets 0. In the symmetric mixed equilibrium each of $n$ players declines with probability $q$ that makes them indifferent:

Bigger groups make each person less likely to act and make total failure more likely. The Kitty Genovese murder (1964) is often cited as the example, but later investigations found the original account of dozens of passive witnesses was inaccurate. Fix: assign one named owner (on-call rotas, "you, call an ambulance", a directly responsible individual).

Sequential games and commitment

A game tree (extensive form) shows who moves when, what they know, and payoffs at each end. Backward induction: start at the last decisions, pick each mover's best action, replace that node with its payoff, and work back to the root. A strategy profile is a subgame-perfect equilibrium (Selten) if it is a Nash equilibrium in every subgame, including those never reached. That rules out threats nobody would carry out.

The same game in normal form (payoffs: entrant, incumbent):

There are two Nash equilibria. (Stay out, fight) is sustained by a threat that costs the incumbent nothing because it is never tested; but if entry happened, fighting would earn 2 instead of 5, so the threat is not credible. The subgame-perfect equilibrium is (enter, accommodate). The 1994 prize press release uses exactly this example to explain Selten's refinement.

Making threats and promises credible

Schelling's central insight, in the 2005 prize summary's words: "a party can strengthen its position by overtly worsening its own options". Commitment works only if it is visible, irreversible and understood by the other side, and it costs flexibility: if the world changes, you are stuck with it.

Chain-store paradox (Selten 1978). A chain faces a potential entrant in each of 20 towns in turn. In the last town, fighting cannot deter anyone, so the chain accommodates; knowing that, fighting in town 19 cannot deter either, and so on back to town 1: backward induction says accommodate everywhere. Yet intuitively a chain should fight early to build a reputation. Kreps and Wilson (1982) and Milgrom and Roberts (1982) resolved it: if entrants think there is even a small chance the incumbent is a "tough type" who likes fighting, a rational incumbent fights early to mimic that type. Reputation needs uncertainty about your type.

First mover or second mover?

The first-mover advantage is weaker than folklore suggests. Lieberman and Montgomery (1988) reviewed the mechanisms and their limits; Golder and Tellis (1993) argued that much apparent pioneer advantage is survivor bias, because failed pioneers are forgotten and later entrants get labeled pioneers. Peter Thiel argues in Zero to One (2014) for being the last mover: the one who dominates a market for the long term (see Peter Thiel).

Repeated games and cooperation

A one-shot prisoner's dilemma ends in mutual defection. Repeat it indefinitely and cooperation can be an equilibrium, because defecting today triggers punishment tomorrow. Axelrod called this the shadow of the future.

Grim trigger: cooperate until the other defects once, then defect forever. Let $\delta$ be the weight on the next round (discount factor times the probability the game continues). Against a grim-trigger partner:

With $T = 5$, $R = 3$, $P = 1$: cooperation is sustainable if $\delta \ge 0.5$. If there is no discounting but a 10% chance each round is the last, $\delta = 0.9$ and cooperation holds comfortably. Against tit-for-tat, a deviator can also alternate defect and cooperate; blocking that needs $\delta \ge \frac{T-R}{R-S} = \frac{2}{3}$ here.

The 2005 prize summary lists the flip side: cooperation is harder with many participants, infrequent interaction, a likely break-up, a short horizon, or actions others cannot clearly observe.

Finite horizon. If everyone knows the game ends after exactly $N$ rounds, backward induction unravels cooperation from the last round back to the first. In practice people cooperate until near the end, and Kreps, Milgrom, Roberts and Wilson (1982) showed that a small doubt about the other's rationality or type is enough to make cooperation rational for most of a finite game. End-game effects are real, though: expect behavior to change when a relationship's end becomes visible (the last months of a contract, a departing employee, a company being sold).

Folk theorem (informal). In an infinitely repeated game with patient enough players, any outcome that gives each player more than they could guarantee alone can be sustained as an equilibrium, cooperation included (formal versions include Friedman 1971 and Fudenberg and Maskin 1986; Aumann's work on repeated games was part of his 2005 prize). The name reflects that the result was widely known before anyone published it. The upside: cooperation is possible. The downside: so is almost everything else, so repetition alone does not predict which outcome you get.

Axelrod's tournaments

Axelrod invited game theorists and others to submit programs for an iterated prisoner's dilemma. The first tournament (reported 1980) had 14 entries playing 200-move matches in a round robin; the second had 62 entrants who had seen the first results. Tit for tat (TFT: cooperate first, then copy the opponent's last move), submitted by Anatol Rapoport, won both. Axelrod attributed its success to four properties:

His advice to players, paraphrased from The Evolution of Cooperation (1984): do not be envious of your partner's score, do not be the first to defect, reciprocate both cooperation and defection, and do not be too clever.

Caveats: TFT's victories depended partly on Axelrod's setup and the other entries. In the 2004–05 twentieth-anniversary tournaments, in the one closest to his design, TFT finished fourteenth out of fifty (Stanford Encyclopedia of Philosophy, "Prisoner's Dilemma"). The robust lessons are the qualitative ones: be nice, reciprocate, forgive, and be legible, with more generosity when noise (misread emails, honest mistakes) is common.

Reputation generalizes repetition: when strangers can see how you treated others (reviews, references, a track record), each interaction borrows the shadow of the future from all the others. Guard it accordingly: reputation is slow to build and fast to lose.

Information: signaling, screening, bluffing

Signaling

A signal is an action that is cheaper for good types than for bad types, so only good types find it worth sending. In Spence's job-market model (1973), education may add nothing to productivity yet still separate workers: if it is costlier for less able workers, there is an equilibrium where able workers get the credential, others do not, and employers pay by credential. The condition for a separating equilibrium is that the wage premium exceeds the high type's cost of the signal but not the low type's.

In biology Zahavi's handicap principle (1975) argues that signals are honest because they are costly: a peacock's tail is credible because a weak bird could not afford it (formalised by Grafen 1990).

Cheap talk (costless, unverifiable messages) can still carry information when interests are aligned; the more they diverge, the less a message can convey (Crawford and Sobel 1982). "Our price is final" is cheap talk unless something makes it costly to back down.

Screening

The uninformed side designs a menu so that types self-select (Rothschild and Stiglitz 1976 for insurance). Examples: insurance deductibles (low-risk buyers choose high deductibles), take-home tests and work samples in hiring, a cheap self-serve tier and a pricey enterprise tier, "pay less for a longer commitment" plans that attract loyal customers.

Bluffing

In games of incomplete information, playing only strong hands strongly is predictable and exploitable, so equilibrium play mixes in bluffs. Poker is the canonical case (von Neumann and Morgenstern analyzed simplified poker in 1944). A bettor wagers $B$ into a pot of $P$. If a fraction $b$ of their bets are bluffs, the caller is indifferent when $b(P + B) = (1 - b)B$, so

A pot-sized bet ($B = P$) should be a bluff one time in three, and the defender should call half the time. The same logic applies to negotiation: if you only ever threaten to walk away when you mean it, your threats are informative and your bluffs are worthless; if you never follow through, neither are your threats.

Bargaining and negotiation

Nash bargaining solution (Nash 1950). Two parties split a surplus; if they fail to agree, they get disagreement payoffs $d_1, d_2$. Under axioms of efficiency, symmetry, independence of irrelevant alternatives and invariance to rescaling utility, the solution maximizes the product of gains:

With money and linear utility, each gets their outside option plus half the surplus above both outside options: $x_i = d_i + \tfrac12(S - d_1 - d_2)$. Splitting $100 when A can get $30 elsewhere and B $10: A gets $30 + 30 = 60$, B gets $10 + 30 = 40$. Improving your outside option is worth half its value in the deal.

Rubinstein alternating offers (1982). Players take turns proposing; each round of delay shrinks the pie by discount factors $\delta_1, \delta_2$. The unique subgame-perfect outcome is immediate agreement, with the first proposer getting

Patience is power: the side that loses less from delay gets more, and moving first matters less than being able to wait. Runway, a strong alternative and a lack of deadline pressure all raise your effective $\delta$.

BATNA and ZOPA

Fisher and Ury's Getting to Yes (1981) introduced the BATNA, the best alternative to a negotiated agreement: what you will do if this deal fails. Your reservation price follows from it. The ZOPA (zone of possible agreement) is the overlap between the two reservation prices; if there is none, no deal should happen.

What moves the result: improving your BATNA (another offer; another candidate), learning theirs, expanding the pie with non-salary terms (equity, start date, title, remote work), and anchoring: first offers pull final agreements toward them (Galinsky and Mussweiler 2001; see cognitive biases). Their four principles (separate the people from the problem, focus on interests not positions, invent options for mutual gain, insist on objective criteria) are ways to turn a fixed-pie split into a larger-pie game.

Auctions, mechanism design and matching

Why truthful bidding is dominant in a Vickrey auction. Your bid only decides whether you win, never what you pay. Bidding above your value $v$ changes the outcome only when the highest rival bid lies between $v$ and your bid: you then win and pay more than $v$, a loss. Bidding below $v$ changes the outcome only when the highest rival bid lies between your bid and $v$: you then lose an auction you would have won at a profit. So bidding $v$ is never worse and sometimes better.

Revenue equivalence (informal; Vickrey 1961, generalized by Myerson 1981 and Riley and Samuelson 1981). With risk-neutral bidders whose private values are independent draws from the same distribution, any format in which the highest-value bidder wins and a zero-value bidder pays nothing yields the same expected revenue. Example: 3 bidders, values uniform on [0, 1]. Second price: expected second-highest value $= \frac{n-1}{n+1} = 0.5$. First price: expected highest value $\frac34$ times shading $\frac23 = 0.5$. Formats differ once those assumptions break (risk aversion, correlated values, collusion, asymmetric bidders).

Winner's curse. When the item has a common value (oil in a tract, a startup's true worth, a spectrum license), the winner is the bidder whose estimate was most optimistic, so naive winners overpay. Capen, Clapp and Campbell (1971) used it to explain oil companies' low returns on offshore lease bids; Wilson formalised it, showing why rational bidders bid below their own estimates. Rule: bid as if your estimate is the highest, because conditional on winning, it is. The correction grows with the number of rival bidders.

Mechanism design is game theory in reverse: choose the rules so that self-interested players with private information produce the outcome you want. Key ideas: incentive compatibility (telling the truth is in each player's interest); the revelation principle (any outcome achievable by some mechanism is achievable by one where players simply report their types truthfully); and hard limits such as Myerson and Satterthwaite (1983): with private valuations there is generally no mechanism that guarantees efficient bilateral trade, voluntary participation and budget balance at once. Hurwicz, Maskin and Myerson shared the 2007 prize. In practice: Milgrom and Wilson (with Preston McAfee) designed the simultaneous multiple round auction, first used by the US Federal Communications Commission in July 1994, when it sold 10 licenses in 47 rounds for $617 million (2020 prize materials).

Matching markets

Some markets have no prices: doctors and hospitals, students and schools, kidney donors and patients. A matching is stable if no pair would rather be with each other than with their assigned partners. Gale and Shapley (1962) proved a stable matching always exists and gave the deferred acceptance algorithm:

1. Each proposer applies to their favorite remaining choice.
2. Each receiver holds the best application so far
   (tentatively) and rejects the rest.
3. Rejected proposers apply to their next choice.
4. Repeat until no one is rejected; then make holds final.

Roth applied it to the matching of new doctors to hospitals, school choice and kidney exchange; Roth and Shapley shared the 2012 prize. Lesson for designers: if a clearinghouse is unstable, participants go around it (early exploding offers, side deals) and it unravels.

Voting and social choice

A Condorcet cycle with three voters:

A beats B 2–1, B beats C 2–1, and C beats A 2–1. Vote A vs B first and the winner then faces C: C wins. Put C vs A first: B wins. Strategic voting is voting against your sincere ranking to avoid a worse outcome (the "wasted vote" logic behind two-party systems under plurality rule). In meetings, the practical lesson is that the order in which options are put to a vote, and which options are on the list, often decide the result.

Behavioral game theory

Models that fit better:

Practical reading: equilibrium analysis is most reliable when stakes are high, players are experienced, the game is simple and repeated, and mistakes are costly (professional penalty takers, spectrum auctions). It is least reliable in novel one-shot situations requiring many steps of reasoning. In between, predict what the actual people will do, not what a perfectly rational stranger would.

Applying it

Business: co-opetition and PARTS

Brandenburger and Nalebuff's HBR article "The Right Game" (1995) and book Co-opetition (1996) argued that business, unlike war and sport, is not only about winning and losing: firms can succeed without others failing. Their value net maps customers, suppliers, competitors and complementors (players whose products make yours more valuable). Their PARTS checklist lists the elements of a game you can change:

Startups

Geopolitics

Deterrence theory is where Schelling's work began. Mutual assured destruction is stable if each side has a secure second-strike capability, so striking first cannot prevent retaliation. Brinkmanship deliberately raises the shared risk of disaster so the other side backs down; Schelling called this "the threat that leaves something to chance", and the 2005 prize summary credits him with showing that "uncertain retaliation is more credible and more efficient than certain retaliation". The Cuban Missile Crisis (October 1962) is commonly framed as chicken, with both sides steering away from nuclear war; historians also stress that it ended in a bargain, including a private US undertaking to remove Jupiter missiles from Turkey, so it was not a pure test of nerve. Treat the framing as a lens, not a history. For tempo and decision cycles in conflict, see the OODA loop.

Changing the game

The single most useful practical lesson: if the equilibrium is bad, stop trying to play it better and change the game so a better outcome becomes the equilibrium.

Common mistakes and critiques

Broader critiques worth knowing: equilibrium concepts often predict many outcomes (the folk theorem is the extreme case); rational-choice assumptions are empirically shaky in novel situations; payoffs are hard to observe, so models can be fitted to any behavior after the fact; and strategic framing can itself erode trust if applied to relationships that were not adversarial.

Strategic analysis template

SITUATION: ______________________________________
 
1. PLAYERS
   Who decides? Who else affects payoffs (regulators,
   complementors, future entrants, the public)?
 
2. ACTIONS AND STRATEGIES
   What can each player do? What is the order of moves?
   Simultaneous or sequential? Can anyone commit first?
 
3. PAYOFFS
   What does each player really value (money, status,
   fairness, reputation, the future)? Rank outcomes for
   each player, not just for me.
 
4. INFORMATION
   What does each know? What is private (costs, resolve,
   alternatives)? What could be signaled or screened?
 
5. REPETITION
   One-shot or repeated? How long is the horizon? Who is
   watching (reputation)? Is the end visible?
 
6. SOLVE
   Dominant strategies? Dominated ones to delete?
   Best responses and Nash equilibria? For sequential
   games, backward induction: which threats are credible?
 
7. DIAGNOSE
   Is the equilibrium efficient? If not, which game is
   it (dilemma, coordination, chicken, commons)?
 
8. CHANGE THE GAME
   Payoffs / players / information / timing /
   commitment / repetition / reputation / scope / rules.
   Which lever is cheapest and most credible?
 
9. ALTERNATIVES
   My BATNA: ______   Their BATNA: ______   ZOPA: ______
 
10. CHECK
   How would a real (not perfectly rational) opponent
   play? What if I'm wrong about their payoffs?

Checklist

• I have listed every player whose choices change the outcome, not just the obvious rival
• I have written payoffs from each player's point of view, including non-financial ones
• I know whether the game is one-shot or repeated, and how visible my behavior is to others
• I have checked for dominant and dominated strategies before anything more complex
• I have found the equilibria, and asked which one is likely and why (focal point, history, first move)
• Every threat or promise I rely on is credible: I would carry it out when the time comes
• I know my BATNA and have estimated theirs
• If the equilibrium is bad, I have looked for a lever to change the game rather than playing harder
• I have sanity-checked the prediction against how real people behave in similar games
• I have a walk-away limit for any escalation or all-pay contest

References

• Stanford Encyclopedia of Philosophy: Game Theory (opens in a new tab): rigorous, readable survey of concepts and foundations
• Stanford Encyclopedia of Philosophy: Prisoner's Dilemma (opens in a new tab): history (Flood, Dresher, Tucker), iterated play, Axelrod and later tournaments
• Yale Open Courses: ECON 159 Game Theory (Ben Polak, 2007) (opens in a new tab): free lecture course, the best video introduction
• John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, 1944): the founding text
• Thomas C. Schelling, The Strategy of Conflict (Harvard University Press, 1960): commitment, credibility, focal points, deterrence
• Avinash Dixit and Barry Nalebuff, Thinking Strategically (W. W. Norton, 1991) and The Art of Strategy (W. W. Norton, 2008): the best non-technical books on applied game theory
• Martin J. Osborne, An Introduction to Game Theory (Oxford University Press, 2004): rigorous undergraduate textbook
• Robert Axelrod, The Evolution of Cooperation (Basic Books, 1984): the iterated prisoner's dilemma tournaments and tit for tat
• Axelrod (1980), Effective Choice in the Prisoner's Dilemma (opens in a new tab), Journal of Conflict Resolution 24(1): the first tournament
• R. Duncan Luce and Howard Raiffa, Games and Decisions (Wiley, 1957): classic text; first printed use of "prisoner's dilemma"
• Elinor Ostrom, Governing the Commons (Cambridge University Press, 1990): the eight design principles
• Roger Fisher and William Ury, Getting to Yes (Houghton Mifflin, 1981): principled negotiation and BATNA
• Adam Brandenburger and Barry Nalebuff, Co-opetition (1996): value net and PARTS
• Brandenburger and Nalebuff (1995), The Right Game: Use Game Theory to Shape Strategy (opens in a new tab), Harvard Business Review: the PARTS framework
• Nobel Prize 1994 press release (opens in a new tab): Nash, Harsanyi, Selten; history and the entry-deterrence example
• Nobel Prize 1996 press release (opens in a new tab): Vickrey auctions and the revelation principle
• Nobel Prize 2005 press release (opens in a new tab): Schelling on commitment, Aumann on repeated games
• Nobel Prize 2007 press release (opens in a new tab): mechanism design
• Nobel Prize 2012 press release (opens in a new tab): stable matching and market design
• Nobel Prize 2020 popular information (opens in a new tab): auction formats, the winner's curse, the 1994 FCC auction
• Nash (1951), Non-Cooperative Games (opens in a new tab), Annals of Mathematics 54: existence of equilibrium
• Nash (1950), The Bargaining Problem (opens in a new tab), Econometrica 18: the Nash bargaining solution
• Rubinstein (1982), Perfect Equilibrium in a Bargaining Model (opens in a new tab), Econometrica 50: alternating offers
• Maynard Smith and Price (1973), The Logic of Animal Conflict (opens in a new tab), Nature 246: evolutionarily stable strategies
• Hardin (1968), The Tragedy of the Commons (opens in a new tab), Science 162
• Chiappori, Levitt and Groseclose (2002), Testing Mixed-Strategy Equilibria: Penalty Kicks in Soccer (opens in a new tab), American Economic Review 92(4)
• Nagel (1995), Unraveling in Guessing Games (opens in a new tab), American Economic Review 85(5): the ⅔ beauty contest
• Güth, Schmittberger and Schwarze (1982), An Experimental Analysis of Ultimatum Bargaining (opens in a new tab), Journal of Economic Behavior and Organization 3
• Güth and Kocher (2013), More than Thirty Years of Ultimatum Bargaining Experiments (opens in a new tab): survey of the evidence
• Fehr and Gächter (2000), Cooperation and Punishment in Public Goods Experiments (opens in a new tab), American Economic Review 90(4)
• Shubik (1971), The Dollar Auction Game (opens in a new tab), Journal of Conflict Resolution 15(1)
• Selten (1978), The Chain Store Paradox (opens in a new tab), Theory and Decision 9
• Diekmann (1985), Volunteer's Dilemma (opens in a new tab), Journal of Conflict Resolution 29(4)
• Capra, Goeree, Gomez and Holt (1999), Anomalous Behavior in a Traveler's Dilemma? (opens in a new tab), American Economic Review 89(3)
• McKelvey and Palfrey (1992), An Experimental Study of the Centipede Game (opens in a new tab), Econometrica 60
• Nowak and Sigmund (1993), A Strategy of Win-Stay, Lose-Shift that Outperforms Tit-for-Tat (opens in a new tab), Nature 364
• Spence (1973), Job Market Signaling (opens in a new tab), Quarterly Journal of Economics 87(3)
• Crawford and Sobel (1982), Strategic Information Transmission (opens in a new tab), Econometrica 50: cheap talk
• Vickrey (1961), Counterspeculation, Auctions, and Competitive Sealed Tenders (opens in a new tab), Journal of Finance 16
• Capen, Clapp and Campbell (1971), Competitive Bidding in High-Risk Situations (opens in a new tab), Journal of Petroleum Technology 23: the winner's curse
• Gale and Shapley (1962), College Admissions and the Stability of Marriage (opens in a new tab), American Mathematical Monthly 69
• Camerer, Ho and Chong (2004), A Cognitive Hierarchy Model of Games (opens in a new tab), Quarterly Journal of Economics 119(3)
• McKelvey and Palfrey (1995), Quantal Response Equilibria for Normal Form Games (opens in a new tab), Games and Economic Behavior 10
• Galinsky and Mussweiler (2001), First Offers as Anchors (opens in a new tab), Journal of Personality and Social Psychology 81(4)
• Golder and Tellis (1993), Pioneer Advantage: Marketing Logic or Marketing Legend? (opens in a new tab), Journal of Marketing Research 30
• Microeconomics: oligopoly models, market failure, adverse selection